LMFDB - Embedded newform 3680.2.i.b.1471.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1471,3680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3680.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.i (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.12
Character	$\chi$	$=$	3680.1471
Dual form			3680.2.i.b.1471.37

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.89612i q^{3} +1.00000i q^{5} +4.39085 q^{7} -0.595278 q^{9} +O(q^{10})$	$q-1.89612i q^{3} +1.00000i q^{5} +4.39085 q^{7} -0.595278 q^{9} +5.84225 q^{11} +4.29622 q^{13} +1.89612 q^{15} -6.96280i q^{17} -2.11167 q^{19} -8.32558i q^{21} +(-4.51980 - 1.60356i) q^{23} -1.00000 q^{25} -4.55965i q^{27} -3.36258 q^{29} +8.00785i q^{31} -11.0776i q^{33} +4.39085i q^{35} -0.681746i q^{37} -8.14616i q^{39} +7.33008 q^{41} +2.93077 q^{43} -0.595278i q^{45} +4.08162i q^{47} +12.2795 q^{49} -13.2023 q^{51} -4.41170i q^{53} +5.84225i q^{55} +4.00398i q^{57} -8.67203i q^{59} +5.30759i q^{61} -2.61377 q^{63} +4.29622i q^{65} +8.04944 q^{67} +(-3.04054 + 8.57009i) q^{69} +11.7974i q^{71} -15.2094 q^{73} +1.89612i q^{75} +25.6524 q^{77} -15.3745 q^{79} -10.4315 q^{81} -6.75102 q^{83} +6.96280 q^{85} +6.37587i q^{87} +11.8984i q^{89} +18.8640 q^{91} +15.1839 q^{93} -2.11167i q^{95} +15.2966i q^{97} -3.47776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 8 q^{7} - 48 q^{9}+O(q^{10}) $ 48 * q + 8 * q^7 - 48 * q^9	$ 48 q + 8 q^{7} - 48 q^{9} + 16 q^{11} + 20 q^{23} - 48 q^{25} - 8 q^{29} + 8 q^{41} + 56 q^{49} - 24 q^{51} - 120 q^{63} - 32 q^{67} - 20 q^{69} + 32 q^{77} + 72 q^{81} + 64 q^{83} + 40 q^{91} - 32 q^{93} - 144 q^{99}+O(q^{100}) $ 48 * q + 8 * q^7 - 48 * q^9 + 16 * q^11 + 20 * q^23 - 48 * q^25 - 8 * q^29 + 8 * q^41 + 56 * q^49 - 24 * q^51 - 120 * q^63 - 32 * q^67 - 20 * q^69 + 32 * q^77 + 72 * q^81 + 64 * q^83 + 40 * q^91 - 32 * q^93 - 144 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3680\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1381$	$3041$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.89612i		−	1.09473i	−0.836895	−	0.547363i	$-0.815632\pi$
$3$		−	1.89612i		−	1.09473i	0.836895	−	0.547363i	$-0.184368\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	4.39085			1.65958			0.829792	−	0.558073i	$-0.188459\pi$
$7$	4.39085			1.65958			0.829792	+	0.558073i	$0.188459\pi$
$8$		0			0
$9$	−0.595278			−0.198426
$10$		0			0
$11$	5.84225			1.76150			0.880752	−	0.473578i	$-0.157038\pi$
$11$	5.84225			1.76150			0.880752	+	0.473578i	$0.157038\pi$
$12$		0			0
$13$	4.29622			1.19156			0.595778	−	0.803149i	$-0.296844\pi$
$13$	4.29622			1.19156			0.595778	+	0.803149i	$0.296844\pi$
$14$		0			0
$15$	1.89612			0.489577
$16$		0			0
$17$		−	6.96280i		−	1.68873i	−0.535771	−	0.844363i	$-0.679979\pi$
$17$		−	6.96280i		−	1.68873i	0.535771	−	0.844363i	$-0.320021\pi$
$18$		0			0
$19$	−2.11167			−0.484449			−0.242225	−	0.970220i	$-0.577877\pi$
$19$	−2.11167			−0.484449			−0.242225	+	0.970220i	$0.577877\pi$
$20$		0			0
$21$		−	8.32558i		−	1.81679i
$22$		0			0
$23$	−4.51980	−	1.60356i	−0.942444	−	0.334365i
$23$	−4.51980	−	1.60356i	−0.942444	−	0.334365i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	4.55965i		−	0.877504i
$28$		0			0
$29$	−3.36258			−0.624416			−0.312208	−	0.950014i	$-0.601069\pi$
$29$	−3.36258			−0.624416			−0.312208	+	0.950014i	$0.601069\pi$
$30$		0			0
$31$			8.00785i			1.43825i	0.694880	+	0.719126i	$0.255458\pi$
$31$			8.00785i			1.43825i	−0.694880	+	0.719126i	$0.744542\pi$
$32$		0			0
$33$		−	11.0776i		−	1.92836i
$34$		0			0
$35$			4.39085i			0.742189i
$36$		0			0
$37$		−	0.681746i		−	0.112078i	−0.998429	−	0.0560392i	$-0.982153\pi$
$37$		−	0.681746i		−	0.112078i	0.998429	−	0.0560392i	$-0.0178472\pi$
$38$		0			0
$39$		−	8.14616i		−	1.30443i
$40$		0			0
$41$	7.33008			1.14477			0.572383	−	0.819986i	$-0.306019\pi$
$41$	7.33008			1.14477			0.572383	+	0.819986i	$0.306019\pi$
$42$		0			0
$43$	2.93077			0.446938			0.223469	−	0.974711i	$-0.428262\pi$
$43$	2.93077			0.446938			0.223469	+	0.974711i	$0.428262\pi$
$44$		0			0
$45$		−	0.595278i		−	0.0887387i
$46$		0			0
$47$			4.08162i			0.595366i	0.954665	+	0.297683i	$0.0962138\pi$
$47$			4.08162i			0.595366i	−0.954665	+	0.297683i	$0.903786\pi$
$48$		0			0
$49$	12.2795			1.75422
$50$		0			0
$51$	−13.2023			−1.84869
$52$		0			0
$53$		−	4.41170i		−	0.605994i	−0.952992	−	0.302997i	$-0.902013\pi$
$53$		−	4.41170i		−	0.605994i	0.952992	−	0.302997i	$-0.0979872\pi$
$54$		0			0
$55$			5.84225i			0.787768i
$56$		0			0
$57$			4.00398i			0.530339i
$58$		0			0
$59$		−	8.67203i		−	1.12900i	−0.825432	−	0.564501i	$-0.809069\pi$
$59$		−	8.67203i		−	1.12900i	0.825432	−	0.564501i	$-0.190931\pi$
$60$		0			0
$61$			5.30759i			0.679568i	0.940504	+	0.339784i	$0.110354\pi$
$61$			5.30759i			0.679568i	−0.940504	+	0.339784i	$0.889646\pi$
$62$		0			0
$63$	−2.61377			−0.329304
$64$		0			0
$65$			4.29622i			0.532880i
$66$		0			0
$67$	8.04944			0.983396			0.491698	−	0.870766i	$-0.336376\pi$
$67$	8.04944			0.983396			0.491698	+	0.870766i	$0.336376\pi$
$68$		0			0
$69$	−3.04054	+	8.57009i	−0.366038	+	1.03172i
$70$		0			0
$71$			11.7974i			1.40009i	0.714097	+	0.700047i	$0.246838\pi$
$71$			11.7974i			1.40009i	−0.714097	+	0.700047i	$0.753162\pi$
$72$		0			0
$73$	−15.2094			−1.78012			−0.890062	−	0.455839i	$-0.849339\pi$
$73$	−15.2094			−1.78012			−0.890062	+	0.455839i	$0.849339\pi$
$74$		0			0
$75$			1.89612i			0.218945i
$76$		0			0
$77$	25.6524			2.92336
$78$		0			0
$79$	−15.3745			−1.72977			−0.864883	−	0.501973i	$-0.832608\pi$
$79$	−15.3745			−1.72977			−0.864883	+	0.501973i	$0.832608\pi$
$80$		0			0
$81$	−10.4315			−1.15905
$82$		0			0
$83$	−6.75102			−0.741020			−0.370510	−	0.928828i	$-0.620817\pi$
$83$	−6.75102			−0.741020			−0.370510	+	0.928828i	$0.620817\pi$
$84$		0			0
$85$	6.96280			0.755222
$86$		0			0
$87$			6.37587i			0.683565i
$88$		0			0
$89$			11.8984i			1.26123i	0.776096	+	0.630614i	$0.217197\pi$
$89$			11.8984i			1.26123i	−0.776096	+	0.630614i	$0.782803\pi$
$90$		0			0
$91$	18.8640			1.97749
$92$		0			0
$93$	15.1839			1.57449
$94$		0			0
$95$		−	2.11167i		−	0.216652i
$96$		0			0
$97$			15.2966i			1.55313i	0.630036	+	0.776566i	$0.283040\pi$
$97$			15.2966i			1.55313i	−0.630036	+	0.776566i	$0.716960\pi$
$98$		0			0
$99$	−3.47776			−0.349528
$100$		0			0
$101$	18.5549			1.84628			0.923141	−	0.384461i	$-0.125613\pi$
$101$	18.5549			1.84628			0.923141	+	0.384461i	$0.125613\pi$
$102$		0			0
$103$	−4.39586			−0.433137			−0.216569	−	0.976267i	$-0.569487\pi$
$103$	−4.39586			−0.433137			−0.216569	+	0.976267i	$0.569487\pi$
$104$		0			0
$105$	8.32558			0.812493
$106$		0			0
$107$	−20.2921			−1.96171			−0.980855	−	0.194742i	$-0.937613\pi$
$107$	−20.2921			−1.96171			−0.980855	+	0.194742i	$0.937613\pi$
$108$		0			0
$109$			1.99737i			0.191313i	0.995414	+	0.0956565i	$0.0304950\pi$
$109$			1.99737i			0.191313i	−0.995414	+	0.0956565i	$0.969505\pi$
$110$		0			0
$111$	−1.29267			−0.122695
$112$		0			0
$113$		−	7.68113i		−	0.722580i	−0.932454	−	0.361290i	$-0.882337\pi$
$113$		−	7.68113i		−	0.722580i	0.932454	−	0.361290i	$-0.117663\pi$
$114$		0			0
$115$	1.60356	−	4.51980i	0.149532	−	0.421474i
$116$		0			0
$117$	−2.55744			−0.236436
$118$		0			0
$119$		−	30.5726i		−	2.80258i
$120$		0			0
$121$	23.1318			2.10289
$122$		0			0
$123$		−	13.8987i		−	1.25321i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			7.43654i			0.659886i	0.944001	+	0.329943i	$0.107030\pi$
$127$			7.43654i			0.659886i	−0.944001	+	0.329943i	$0.892970\pi$
$128$		0			0
$129$		−	5.55709i		−	0.489275i
$130$		0			0
$131$		−	13.7753i		−	1.20355i	−0.798665	−	0.601776i	$-0.794460\pi$
$131$		−	13.7753i		−	1.20355i	0.798665	−	0.601776i	$-0.205540\pi$
$132$		0			0
$133$	−9.27200			−0.803984
$134$		0			0
$135$	4.55965			0.392432
$136$		0			0
$137$			0.337739i			0.0288550i	0.999896	+	0.0144275i	$0.00459258\pi$
$137$			0.337739i			0.0288550i	−0.999896	+	0.0144275i	$0.995407\pi$
$138$		0			0
$139$			9.90342i			0.839997i	0.907525	+	0.419999i	$0.137969\pi$
$139$			9.90342i			0.839997i	−0.907525	+	0.419999i	$0.862031\pi$
$140$		0			0
$141$	7.73925			0.651763
$142$		0			0
$143$	25.0996			2.09893
$144$		0			0
$145$		−	3.36258i		−	0.279247i
$146$		0			0
$147$		−	23.2835i		−	1.92039i
$148$		0			0
$149$			3.21838i			0.263660i	0.991272	+	0.131830i	$0.0420853\pi$
$149$			3.21838i			0.263660i	−0.991272	+	0.131830i	$0.957915\pi$
$150$		0			0
$151$			13.7808i			1.12147i	0.827997	+	0.560733i	$0.189480\pi$
$151$			13.7808i			1.12147i	−0.827997	+	0.560733i	$0.810520\pi$
$152$		0			0
$153$			4.14480i			0.335087i
$154$		0			0
$155$	−8.00785			−0.643206
$156$		0			0
$157$		−	17.1598i		−	1.36950i	−0.728778	−	0.684750i	$-0.759911\pi$
$157$		−	17.1598i		−	1.36950i	0.728778	−	0.684750i	$-0.240089\pi$
$158$		0			0
$159$	−8.36512			−0.663397
$160$		0			0
$161$	−19.8458	−	7.04097i	−1.56406	−	0.554907i
$162$		0			0
$163$		−	9.39428i		−	0.735817i	−0.929862	−	0.367908i	$-0.880074\pi$
$163$		−	9.39428i		−	0.735817i	0.929862	−	0.367908i	$-0.119926\pi$
$164$		0			0
$165$	11.0776			0.862391
$166$		0			0
$167$		−	14.4213i		−	1.11595i	−0.829857	−	0.557976i	$-0.811578\pi$
$167$		−	14.4213i		−	1.11595i	0.829857	−	0.557976i	$-0.188422\pi$
$168$		0			0
$169$	5.45750			0.419808
$170$		0			0
$171$	1.25703			0.0961273
$172$		0			0
$173$	−3.26360			−0.248127			−0.124063	−	0.992274i	$-0.539593\pi$
$173$	−3.26360			−0.248127			−0.124063	+	0.992274i	$0.539593\pi$
$174$		0			0
$175$	−4.39085			−0.331917
$176$		0			0
$177$	−16.4432			−1.23595
$178$		0			0
$179$			22.9296i			1.71384i	0.515452	+	0.856919i	$0.327624\pi$
$179$			22.9296i			1.71384i	−0.515452	+	0.856919i	$0.672376\pi$
$180$		0			0
$181$		−	9.39717i		−	0.698486i	−0.937032	−	0.349243i	$-0.886439\pi$
$181$		−	9.39717i		−	0.698486i	0.937032	−	0.349243i	$-0.113561\pi$
$182$		0			0
$183$	10.0638			0.743941
$184$		0			0
$185$	0.681746			0.0501230
$186$		0			0
$187$		−	40.6784i		−	2.97470i
$188$		0			0
$189$		−	20.0207i		−	1.45629i
$190$		0			0
$191$	4.96782			0.359459			0.179729	−	0.983716i	$-0.442478\pi$
$191$	4.96782			0.359459			0.179729	+	0.983716i	$0.442478\pi$
$192$		0			0
$193$	−16.7773			−1.20766			−0.603828	−	0.797115i	$-0.706358\pi$
$193$	−16.7773			−1.20766			−0.603828	+	0.797115i	$0.706358\pi$
$194$		0			0
$195$	8.14616			0.583358
$196$		0			0
$197$	−0.153918			−0.0109662			−0.00548311	−	0.999985i	$-0.501745\pi$
$197$	−0.153918			−0.0109662			−0.00548311	+	0.999985i	$0.501745\pi$
$198$		0			0
$199$	−20.4354			−1.44863			−0.724313	−	0.689471i	$-0.757843\pi$
$199$	−20.4354			−1.44863			−0.724313	+	0.689471i	$0.757843\pi$
$200$		0			0
$201$		−	15.2627i		−	1.07655i
$202$		0			0
$203$	−14.7646			−1.03627
$204$		0			0
$205$			7.33008i			0.511955i
$206$		0			0
$207$	2.69054	+	0.954562i	0.187005	+	0.0663466i
$208$		0			0
$209$	−12.3369			−0.853359
$210$		0			0
$211$		−	15.0613i		−	1.03686i	−0.855119	−	0.518431i	$-0.826516\pi$
$211$		−	15.0613i		−	1.03686i	0.855119	−	0.518431i	$-0.173484\pi$
$212$		0			0
$213$	22.3693			1.53272
$214$		0			0
$215$			2.93077i			0.199877i
$216$		0			0
$217$			35.1612i			2.38690i
$218$		0			0
$219$			28.8389i			1.94875i
$220$		0			0
$221$		−	29.9137i		−	2.01221i
$222$		0			0
$223$			6.22787i			0.417049i	0.978017	+	0.208525i	$0.0668661\pi$
$223$			6.22787i			0.417049i	−0.978017	+	0.208525i	$0.933134\pi$
$224$		0			0
$225$	0.595278			0.0396852
$226$		0			0
$227$	17.0534			1.13187			0.565937	−	0.824449i	$-0.308515\pi$
$227$	17.0534			1.13187			0.565937	+	0.824449i	$0.308515\pi$
$228$		0			0
$229$		−	13.3789i		−	0.884100i	−0.896990	−	0.442050i	$-0.854251\pi$
$229$		−	13.3789i		−	0.884100i	0.896990	−	0.442050i	$-0.145749\pi$
$230$		0			0
$231$		−	48.6401i		−	3.20028i
$232$		0			0
$233$	20.7178			1.35727			0.678634	−	0.734476i	$-0.262572\pi$
$233$	20.7178			1.35727			0.678634	+	0.734476i	$0.262572\pi$
$234$		0			0
$235$	−4.08162			−0.266256
$236$		0			0
$237$			29.1519i			1.89362i
$238$		0			0
$239$			14.5963i			0.944156i	0.881557	+	0.472078i	$0.156496\pi$
$239$			14.5963i			0.944156i	−0.881557	+	0.472078i	$0.843504\pi$
$240$		0			0
$241$			3.07646i			0.198172i	0.995079	+	0.0990862i	$0.0315919\pi$
$241$			3.07646i			0.198172i	−0.995079	+	0.0990862i	$0.968408\pi$
$242$		0			0
$243$			6.10041i			0.391342i
$244$		0			0
$245$			12.2795i			0.784511i
$246$		0			0
$247$	−9.07218			−0.577249
$248$		0			0
$249$			12.8008i			0.811215i
$250$		0			0
$251$	−30.2599			−1.90998			−0.954992	−	0.296630i	$-0.904137\pi$
$251$	−30.2599			−1.90998			−0.954992	+	0.296630i	$0.904137\pi$
$252$		0			0
$253$	−26.4058	−	9.36838i	−1.66012	−	0.588985i
$254$		0			0
$255$		−	13.2023i		−	0.826761i
$256$		0			0
$257$	−3.94260			−0.245933			−0.122966	−	0.992411i	$-0.539241\pi$
$257$	−3.94260			−0.245933			−0.122966	+	0.992411i	$0.539241\pi$
$258$		0			0
$259$		−	2.99344i		−	0.186003i
$260$		0			0
$261$	2.00167			0.123900
$262$		0			0
$263$	15.7871			0.973476			0.486738	−	0.873548i	$-0.338187\pi$
$263$	15.7871			0.973476			0.486738	+	0.873548i	$0.338187\pi$
$264$		0			0
$265$	4.41170			0.271009
$266$		0			0
$267$	22.5608			1.38070
$268$		0			0
$269$	9.19070			0.560367			0.280184	−	0.959946i	$-0.409605\pi$
$269$	9.19070			0.560367			0.280184	+	0.959946i	$0.409605\pi$
$270$		0			0
$271$			8.37345i			0.508651i	0.967119	+	0.254325i	$0.0818534\pi$
$271$			8.37345i			0.508651i	−0.967119	+	0.254325i	$0.918147\pi$
$272$		0			0
$273$		−	35.7685i		−	2.16481i
$274$		0			0
$275$	−5.84225			−0.352301
$276$		0			0
$277$	−0.0425692			−0.00255774			−0.00127887	−	0.999999i	$-0.500407\pi$
$277$	−0.0425692			−0.00255774			−0.00127887	+	0.999999i	$0.500407\pi$
$278$		0			0
$279$		−	4.76689i		−	0.285386i
$280$		0			0
$281$		−	4.26122i		−	0.254203i	−0.991890	−	0.127101i	$-0.959433\pi$
$281$		−	4.26122i		−	0.254203i	0.991890	−	0.127101i	$-0.0405674\pi$
$282$		0			0
$283$	−18.3971			−1.09359			−0.546797	−	0.837265i	$-0.684153\pi$
$283$	−18.3971			−1.09359			−0.546797	+	0.837265i	$0.684153\pi$
$284$		0			0
$285$	−4.00398			−0.237175
$286$		0			0
$287$	32.1853			1.89984
$288$		0			0
$289$	−31.4806			−1.85180
$290$		0			0
$291$	29.0042			1.70025
$292$		0			0
$293$			21.8096i			1.27413i	0.770811	+	0.637064i	$0.219851\pi$
$293$			21.8096i			1.27413i	−0.770811	+	0.637064i	$0.780149\pi$
$294$		0			0
$295$	8.67203			0.504905
$296$		0			0
$297$		−	26.6386i		−	1.54573i
$298$		0			0
$299$	−19.4181	−	6.88923i	−1.12298	−	0.398415i
$300$		0			0
$301$	12.8686			0.741731
$302$		0			0
$303$		−	35.1824i		−	2.02117i
$304$		0			0
$305$	−5.30759			−0.303912
$306$		0			0
$307$			12.4213i			0.708923i	0.935071	+	0.354461i	$0.115336\pi$
$307$			12.4213i			0.708923i	−0.935071	+	0.354461i	$0.884664\pi$
$308$		0			0
$309$			8.33509i			0.474167i
$310$		0			0
$311$		−	15.8348i		−	0.897911i	−0.893554	−	0.448955i	$-0.851796\pi$
$311$		−	15.8348i		−	0.897911i	0.893554	−	0.448955i	$-0.148204\pi$
$312$		0			0
$313$		−	1.94658i		−	0.110027i	−0.998486	−	0.0550137i	$-0.982480\pi$
$313$		−	1.94658i		−	0.110027i	0.998486	−	0.0550137i	$-0.0175202\pi$
$314$		0			0
$315$		−	2.61377i		−	0.147269i
$316$		0			0
$317$	18.1263			1.01808			0.509038	−	0.860744i	$-0.330001\pi$
$317$	18.1263			1.01808			0.509038	+	0.860744i	$0.330001\pi$
$318$		0			0
$319$	−19.6450			−1.09991
$320$		0			0
$321$			38.4762i			2.14753i
$322$		0			0
$323$			14.7031i			0.818103i
$324$		0			0
$325$	−4.29622			−0.238311
$326$		0			0
$327$	3.78725			0.209435
$328$		0			0
$329$			17.9218i			0.988060i
$330$		0			0
$331$			30.6712i			1.68584i	0.538037	+	0.842921i	$0.319166\pi$
$331$			30.6712i			1.68584i	−0.538037	+	0.842921i	$0.680834\pi$
$332$		0			0
$333$			0.405828i			0.0222392i
$334$		0			0
$335$			8.04944i			0.439788i
$336$		0			0
$337$		−	5.80653i		−	0.316302i	−0.987415	−	0.158151i	$-0.949447\pi$
$337$		−	5.80653i		−	0.316302i	0.987415	−	0.158151i	$-0.0505532\pi$
$338$		0			0
$339$	−14.5644			−0.791027
$340$		0			0
$341$			46.7838i			2.53349i
$342$		0			0
$343$	23.1816			1.25169
$344$		0			0
$345$	−8.57009	−	3.04054i	−0.461398	−	0.163697i
$346$		0			0
$347$		−	29.1918i		−	1.56710i	−0.621331	−	0.783548i	$-0.713408\pi$
$347$		−	29.1918i		−	1.56710i	0.621331	−	0.783548i	$-0.286592\pi$
$348$		0			0
$349$	25.6297			1.37192			0.685962	−	0.727638i	$-0.259382\pi$
$349$	25.6297			1.37192			0.685962	+	0.727638i	$0.259382\pi$
$350$		0			0
$351$		−	19.5892i		−	1.04560i
$352$		0			0
$353$	22.7829			1.21261			0.606306	−	0.795232i	$-0.292651\pi$
$353$	22.7829			1.21261			0.606306	+	0.795232i	$0.292651\pi$
$354$		0			0
$355$	−11.7974			−0.626141
$356$		0			0
$357$	−57.9693			−3.06806
$358$		0			0
$359$	−7.77491			−0.410344			−0.205172	−	0.978726i	$-0.565775\pi$
$359$	−7.77491			−0.410344			−0.205172	+	0.978726i	$0.565775\pi$
$360$		0			0
$361$	−14.5409			−0.765309
$362$		0			0
$363$		−	43.8608i		−	2.30209i
$364$		0			0
$365$		−	15.2094i		−	0.796096i
$366$		0			0
$367$	−12.5553			−0.655380			−0.327690	−	0.944785i	$-0.606270\pi$
$367$	−12.5553			−0.655380			−0.327690	+	0.944785i	$0.606270\pi$
$368$		0			0
$369$	−4.36343			−0.227151
$370$		0			0
$371$		−	19.3711i		−	1.00570i
$372$		0			0
$373$		−	9.87075i		−	0.511088i	−0.966797	−	0.255544i	$-0.917745\pi$
$373$		−	9.87075i		−	0.511088i	0.966797	−	0.255544i	$-0.0822546\pi$
$374$		0			0
$375$	−1.89612			−0.0979153
$376$		0			0
$377$	−14.4464			−0.744027
$378$		0			0
$379$	−16.9501			−0.870666			−0.435333	−	0.900269i	$-0.643369\pi$
$379$	−16.9501			−0.870666			−0.435333	+	0.900269i	$0.643369\pi$
$380$		0			0
$381$	14.1006			0.722395
$382$		0			0
$383$	−4.97781			−0.254354			−0.127177	−	0.991880i	$-0.540592\pi$
$383$	−4.97781			−0.254354			−0.127177	+	0.991880i	$0.540592\pi$
$384$		0			0
$385$			25.6524i			1.30737i
$386$		0			0
$387$	−1.74462			−0.0886840
$388$		0			0
$389$		−	30.4334i		−	1.54304i	−0.636208	−	0.771518i	$-0.719498\pi$
$389$		−	30.4334i		−	1.54304i	0.636208	−	0.771518i	$-0.280502\pi$
$390$		0			0
$391$	−11.1652	+	31.4705i	−0.564651	+	1.59153i
$392$		0			0
$393$	−26.1196			−1.31756
$394$		0			0
$395$		−	15.3745i		−	0.773575i
$396$		0			0
$397$	16.7965			0.842992			0.421496	−	0.906830i	$-0.361505\pi$
$397$	16.7965			0.842992			0.421496	+	0.906830i	$0.361505\pi$
$398$		0			0
$399$			17.5808i			0.880143i
$400$		0			0
$401$		−	28.7249i		−	1.43445i	−0.696840	−	0.717226i	$-0.745411\pi$
$401$		−	28.7249i		−	1.43445i	0.696840	−	0.717226i	$-0.254589\pi$
$402$		0			0
$403$			34.4035i			1.71376i
$404$		0			0
$405$		−	10.4315i		−	0.518344i
$406$		0			0
$407$		−	3.98293i		−	0.197426i
$408$		0			0
$409$	13.5190			0.668469			0.334235	−	0.942490i	$-0.391522\pi$
$409$	13.5190			0.668469			0.334235	+	0.942490i	$0.391522\pi$
$410$		0			0
$411$	0.640395			0.0315884
$412$		0			0
$413$		−	38.0775i		−	1.87367i
$414$		0			0
$415$		−	6.75102i		−	0.331394i
$416$		0			0
$417$	18.7781			0.919567
$418$		0			0
$419$	−6.10046			−0.298027			−0.149014	−	0.988835i	$-0.547610\pi$
$419$	−6.10046			−0.298027			−0.149014	+	0.988835i	$0.547610\pi$
$420$		0			0
$421$		−	11.4005i		−	0.555629i	−0.960635	−	0.277814i	$-0.910390\pi$
$421$		−	11.4005i		−	0.555629i	0.960635	−	0.277814i	$-0.0896100\pi$
$422$		0			0
$423$		−	2.42970i		−	0.118136i
$424$		0			0
$425$			6.96280i			0.337745i
$426$		0			0
$427$			23.3048i			1.12780i
$428$		0			0
$429$		−	47.5918i		−	2.29776i
$430$		0			0
$431$	8.72150			0.420100			0.210050	−	0.977691i	$-0.432637\pi$
$431$	8.72150			0.420100			0.210050	+	0.977691i	$0.432637\pi$
$432$		0			0
$433$		−	12.5804i		−	0.604575i	−0.953217	−	0.302287i	$-0.902250\pi$
$433$		−	12.5804i		−	0.604575i	0.953217	−	0.302287i	$-0.0977502\pi$
$434$		0			0
$435$	−6.37587			−0.305699
$436$		0			0
$437$	9.54431	+	3.38618i	0.456566	+	0.161983i
$438$		0			0
$439$			2.56656i			0.122495i	0.998123	+	0.0612477i	$0.0195080\pi$
$439$			2.56656i			0.122495i	−0.998123	+	0.0612477i	$0.980492\pi$
$440$		0			0
$441$	−7.30973			−0.348082
$442$		0			0
$443$			35.7056i			1.69642i	0.529657	+	0.848212i	$0.322321\pi$
$443$			35.7056i			1.69642i	−0.529657	+	0.848212i	$0.677679\pi$
$444$		0			0
$445$	−11.8984			−0.564039
$446$		0			0
$447$	6.10244			0.288635
$448$		0			0
$449$	5.79254			0.273367			0.136684	−	0.990615i	$-0.456356\pi$
$449$	5.79254			0.273367			0.136684	+	0.990615i	$0.456356\pi$
$450$		0			0
$451$	42.8241			2.01651
$452$		0			0
$453$	26.1301			1.22770
$454$		0			0
$455$			18.8640i			0.884360i
$456$		0			0
$457$			5.69935i			0.266604i	0.991075	+	0.133302i	$0.0425581\pi$
$457$			5.69935i			0.266604i	−0.991075	+	0.133302i	$0.957442\pi$
$458$		0			0
$459$	−31.7479			−1.48187
$460$		0			0
$461$	−3.55666			−0.165650			−0.0828250	−	0.996564i	$-0.526394\pi$
$461$	−3.55666			−0.165650			−0.0828250	+	0.996564i	$0.526394\pi$
$462$		0			0
$463$		−	14.0131i		−	0.651243i	−0.945500	−	0.325621i	$-0.894427\pi$
$463$		−	14.0131i		−	0.651243i	0.945500	−	0.325621i	$-0.105573\pi$
$464$		0			0
$465$			15.1839i			0.704134i
$466$		0			0
$467$	20.6197			0.954165			0.477082	−	0.878859i	$-0.341694\pi$
$467$	20.6197			0.954165			0.477082	+	0.878859i	$0.341694\pi$
$468$		0			0
$469$	35.3439			1.63203
$470$		0			0
$471$	−32.5371			−1.49923
$472$		0			0
$473$	17.1223			0.787283
$474$		0			0
$475$	2.11167			0.0968899
$476$		0			0
$477$			2.62619i			0.120245i
$478$		0			0
$479$	−34.2006			−1.56267			−0.781333	−	0.624114i	$-0.785460\pi$
$479$	−34.2006			−1.56267			−0.781333	+	0.624114i	$0.785460\pi$
$480$		0			0
$481$		−	2.92893i		−	0.133548i
$482$		0			0
$483$	−13.3505	+	37.6300i	−0.607471	+	1.71222i
$484$		0			0
$485$	−15.2966			−0.694582
$486$		0			0
$487$			1.73134i			0.0784547i	0.999230	+	0.0392274i	$0.0124897\pi$
$487$			1.73134i			0.0784547i	−0.999230	+	0.0392274i	$0.987510\pi$
$488$		0			0
$489$	−17.8127			−0.805518
$490$		0			0
$491$			3.02913i			0.136703i	0.997661	+	0.0683515i	$0.0217739\pi$
$491$			3.02913i			0.136703i	−0.997661	+	0.0683515i	$0.978226\pi$
$492$		0			0
$493$			23.4130i			1.05447i
$494$		0			0
$495$		−	3.47776i		−	0.156314i
$496$		0			0
$497$			51.8006i			2.32357i
$498$		0			0
$499$			10.7622i			0.481781i	0.970552	+	0.240891i	$0.0774395\pi$
$499$			10.7622i			0.481781i	−0.970552	+	0.240891i	$0.922561\pi$
$500$		0			0
$501$	−27.3445			−1.22166
$502$		0			0
$503$	27.4149			1.22237			0.611185	−	0.791488i	$-0.290693\pi$
$503$	27.4149			1.22237			0.611185	+	0.791488i	$0.290693\pi$
$504$		0			0
$505$			18.5549i			0.825683i
$506$		0			0
$507$		−	10.3481i		−	0.459575i
$508$		0			0
$509$	22.6680			1.00474			0.502371	−	0.864652i	$-0.332461\pi$
$509$	22.6680			1.00474			0.502371	+	0.864652i	$0.332461\pi$
$510$		0			0
$511$	−66.7821			−2.95427
$512$		0			0
$513$			9.62845i			0.425106i
$514$		0			0
$515$		−	4.39586i		−	0.193705i
$516$		0			0
$517$			23.8458i			1.04874i
$518$		0			0
$519$			6.18818i			0.271631i
$520$		0			0
$521$		−	23.5432i		−	1.03145i	−0.856755	−	0.515723i	$-0.827523\pi$
$521$		−	23.5432i		−	1.03145i	0.856755	−	0.515723i	$-0.172477\pi$
$522$		0			0
$523$	11.1787			0.488812			0.244406	−	0.969673i	$-0.421407\pi$
$523$	11.1787			0.488812			0.244406	+	0.969673i	$0.421407\pi$
$524$		0			0
$525$			8.32558i			0.363358i
$526$		0			0
$527$	55.7570			2.42881
$528$		0			0
$529$	17.8572	+	14.4955i	0.776400	+	0.630240i
$530$		0			0
$531$			5.16226i			0.224023i
$532$		0			0
$533$	31.4916			1.36405
$534$		0			0
$535$		−	20.2921i		−	0.877303i
$536$		0			0
$537$	43.4773			1.87618
$538$		0			0
$539$	71.7401			3.09006
$540$		0			0
$541$	−19.8197			−0.852116			−0.426058	−	0.904696i	$-0.640098\pi$
$541$	−19.8197			−0.852116			−0.426058	+	0.904696i	$0.640098\pi$
$542$		0			0
$543$	−17.8182			−0.764651
$544$		0			0
$545$	−1.99737			−0.0855577
$546$		0			0
$547$			7.45444i			0.318729i	0.987220	+	0.159364i	$0.0509445\pi$
$547$			7.45444i			0.318729i	−0.987220	+	0.159364i	$0.949056\pi$
$548$		0			0
$549$		−	3.15949i		−	0.134844i
$550$		0			0
$551$	7.10065			0.302498
$552$		0			0
$553$	−67.5071			−2.87069
$554$		0			0
$555$		−	1.29267i		−	0.0548709i
$556$		0			0
$557$		−	3.36674i		−	0.142654i	−0.997453	−	0.0713268i	$-0.977277\pi$
$557$		−	3.36674i		−	0.142654i	0.997453	−	0.0713268i	$-0.0227233\pi$
$558$		0			0
$559$	12.5912			0.532552
$560$		0			0
$561$	−77.1312			−3.25648
$562$		0			0
$563$	26.8743			1.13262			0.566308	−	0.824194i	$-0.308371\pi$
$563$	26.8743			1.13262			0.566308	+	0.824194i	$0.308371\pi$
$564$		0			0
$565$	7.68113			0.323147
$566$		0			0
$567$	−45.8030			−1.92355
$568$		0			0
$569$			10.1164i			0.424103i	0.977258	+	0.212051i	$0.0680144\pi$
$569$			10.1164i			0.424103i	−0.977258	+	0.212051i	$0.931986\pi$
$570$		0			0
$571$	−5.05950			−0.211734			−0.105867	−	0.994380i	$-0.533762\pi$
$571$	−5.05950			−0.211734			−0.105867	+	0.994380i	$0.533762\pi$
$572$		0			0
$573$		−	9.41958i		−	0.393509i
$574$		0			0
$575$	4.51980	+	1.60356i	0.188489	+	0.0668730i
$576$		0			0
$577$	−13.4936			−0.561748			−0.280874	−	0.959745i	$-0.590624\pi$
$577$	−13.4936			−0.561748			−0.280874	+	0.959745i	$0.590624\pi$
$578$		0			0
$579$			31.8118i			1.32205i
$580$		0			0
$581$	−29.6427			−1.22979
$582$		0			0
$583$		−	25.7742i		−	1.06746i
$584$		0			0
$585$		−	2.55744i		−	0.105737i
$586$		0			0
$587$			24.3309i			1.00424i	0.864797	+	0.502122i	$0.167447\pi$
$587$			24.3309i			1.00424i	−0.864797	+	0.502122i	$0.832553\pi$
$588$		0			0
$589$		−	16.9099i		−	0.696760i
$590$		0			0
$591$			0.291848i			0.0120050i
$592$		0			0
$593$	−0.332256			−0.0136441			−0.00682206	−	0.999977i	$-0.502172\pi$
$593$	−0.332256			−0.0136441			−0.00682206	+	0.999977i	$0.502172\pi$
$594$		0			0
$595$	30.5726			1.25335
$596$		0			0
$597$			38.7480i			1.58585i
$598$		0			0
$599$		−	35.1824i		−	1.43752i	−0.695261	−	0.718758i	$-0.744711\pi$
$599$		−	35.1824i		−	1.43752i	0.695261	−	0.718758i	$-0.255289\pi$
$600$		0			0
$601$	40.2707			1.64268			0.821338	−	0.570442i	$-0.193228\pi$
$601$	40.2707			1.64268			0.821338	+	0.570442i	$0.193228\pi$
$602$		0			0
$603$	−4.79165			−0.195131
$604$		0			0
$605$			23.1318i			0.940443i
$606$		0			0
$607$			43.3538i			1.75968i	0.475272	+	0.879839i	$0.342349\pi$
$607$			43.3538i			1.75968i	−0.475272	+	0.879839i	$0.657651\pi$
$608$		0			0
$609$			27.9955i			1.13443i
$610$		0			0
$611$			17.5355i			0.709412i
$612$		0			0
$613$		−	11.6609i		−	0.470978i	−0.971877	−	0.235489i	$-0.924331\pi$
$613$		−	11.6609i		−	0.470978i	0.971877	−	0.235489i	$-0.0756691\pi$
$614$		0			0
$615$	13.8987			0.560451
$616$		0			0
$617$		−	5.03885i		−	0.202856i	−0.994843	−	0.101428i	$-0.967659\pi$
$617$		−	5.03885i		−	0.202856i	0.994843	−	0.101428i	$-0.0323412\pi$
$618$		0			0
$619$	10.5740			0.425003			0.212502	−	0.977161i	$-0.431839\pi$
$619$	10.5740			0.425003			0.212502	+	0.977161i	$0.431839\pi$
$620$		0			0
$621$	−7.31165	+	20.6087i	−0.293407	+	0.826998i
$622$		0			0
$623$			52.2441i			2.09311i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			23.3922i			0.934195i
$628$		0			0
$629$	−4.74686			−0.189270
$630$		0			0
$631$	−3.80756			−0.151577			−0.0757883	−	0.997124i	$-0.524147\pi$
$631$	−3.80756			−0.151577			−0.0757883	+	0.997124i	$0.524147\pi$
$632$		0			0
$633$	−28.5580			−1.13508
$634$		0			0
$635$	−7.43654			−0.295110
$636$		0			0
$637$	52.7556			2.09025
$638$		0			0
$639$		−	7.02273i		−	0.277815i
$640$		0			0
$641$			15.4843i			0.611592i	0.952097	+	0.305796i	$0.0989225\pi$
$641$			15.4843i			0.611592i	−0.952097	+	0.305796i	$0.901078\pi$
$642$		0			0
$643$	12.4254			0.490009			0.245004	−	0.969522i	$-0.421211\pi$
$643$	12.4254			0.490009			0.245004	+	0.969522i	$0.421211\pi$
$644$		0			0
$645$	5.55709			0.218810
$646$		0			0
$647$			12.0509i			0.473772i	0.971537	+	0.236886i	$0.0761267\pi$
$647$			12.0509i			0.473772i	−0.971537	+	0.236886i	$0.923873\pi$
$648$		0			0
$649$		−	50.6641i		−	1.98874i
$650$		0			0
$651$	66.6700			2.61300
$652$		0			0
$653$	−6.29557			−0.246365			−0.123182	−	0.992384i	$-0.539310\pi$
$653$	−6.29557			−0.246365			−0.123182	+	0.992384i	$0.539310\pi$
$654$		0			0
$655$	13.7753			0.538245
$656$		0			0
$657$	9.05381			0.353223
$658$		0			0
$659$	−33.9302			−1.32173			−0.660866	−	0.750504i	$-0.729811\pi$
$659$	−33.9302			−1.32173			−0.660866	+	0.750504i	$0.729811\pi$
$660$		0			0
$661$			7.12563i			0.277155i	0.990352	+	0.138577i	$0.0442530\pi$
$661$			7.12563i			0.277155i	−0.990352	+	0.138577i	$0.955747\pi$
$662$		0			0
$663$	−56.7200			−2.20282
$664$		0			0
$665$		−	9.27200i		−	0.359553i
$666$		0			0
$667$	15.1982	+	5.39210i	0.588477	+	0.208783i
$668$		0			0
$669$	11.8088			0.456555
$670$		0			0
$671$			31.0083i			1.19706i
$672$		0			0
$673$	−7.41497			−0.285826			−0.142913	−	0.989735i	$-0.545647\pi$
$673$	−7.41497			−0.285826			−0.142913	+	0.989735i	$0.545647\pi$
$674$		0			0
$675$			4.55965i			0.175501i
$676$		0			0
$677$			33.0594i			1.27057i	0.772276	+	0.635287i	$0.219118\pi$
$677$			33.0594i			1.27057i	−0.772276	+	0.635287i	$0.780882\pi$
$678$		0			0
$679$			67.1649i			2.57755i
$680$		0			0
$681$		−	32.3353i		−	1.23909i
$682$		0			0
$683$		−	11.8804i		−	0.454591i	−0.973826	−	0.227295i	$-0.927012\pi$
$683$		−	11.8804i		−	0.454591i	0.973826	−	0.227295i	$-0.0729883\pi$
$684$		0			0
$685$	−0.337739			−0.0129044
$686$		0			0
$687$	−25.3679			−0.967847
$688$		0			0
$689$		−	18.9536i		−	0.722076i
$690$		0			0
$691$			19.9017i			0.757097i	0.925582	+	0.378548i	$0.123577\pi$
$691$			19.9017i			0.757097i	−0.925582	+	0.378548i	$0.876423\pi$
$692$		0			0
$693$	−15.2703			−0.580071
$694$		0			0
$695$	−9.90342			−0.375658
$696$		0			0
$697$		−	51.0379i		−	1.93320i
$698$		0			0
$699$		−	39.2835i		−	1.48584i
$700$		0			0
$701$		−	22.1649i		−	0.837158i	−0.908180	−	0.418579i	$-0.862528\pi$
$701$		−	22.1649i		−	0.837158i	0.908180	−	0.418579i	$-0.137472\pi$
$702$		0			0
$703$			1.43962i			0.0542963i
$704$		0			0
$705$			7.73925i			0.291477i
$706$		0			0
$707$	81.4718			3.06406
$708$		0			0
$709$			7.19666i			0.270276i	0.990827	+	0.135138i	$0.0431478\pi$
$709$			7.19666i			0.270276i	−0.990827	+	0.135138i	$0.956852\pi$
$710$		0			0
$711$	9.15210			0.343230
$712$		0			0
$713$	12.8410	−	36.1939i	0.480901	−	1.35547i
$714$		0			0
$715$			25.0996i			0.938671i
$716$		0			0
$717$	27.6763			1.03359
$718$		0			0
$719$			34.9285i			1.30261i	0.758815	+	0.651306i	$0.225779\pi$
$719$			34.9285i			1.30261i	−0.758815	+	0.651306i	$0.774221\pi$
$720$		0			0
$721$	−19.3016			−0.718828
$722$		0			0
$723$	5.83335			0.216945
$724$		0			0
$725$	3.36258			0.124883
$726$		0			0
$727$	9.78192			0.362791			0.181396	−	0.983410i	$-0.441939\pi$
$727$	9.78192			0.362791			0.181396	+	0.983410i	$0.441939\pi$
$728$		0			0
$729$	−19.7273			−0.730641
$730$		0			0
$731$		−	20.4063i		−	0.754756i
$732$		0			0
$733$			27.4189i			1.01274i	0.862316	+	0.506371i	$0.169013\pi$
$733$			27.4189i			1.01274i	−0.862316	+	0.506371i	$0.830987\pi$
$734$		0			0
$735$	23.2835			0.858825
$736$		0			0
$737$	47.0268			1.73226
$738$		0			0
$739$			6.91545i			0.254389i	0.991878	+	0.127194i	$0.0405972\pi$
$739$			6.91545i			0.254389i	−0.991878	+	0.127194i	$0.959403\pi$
$740$		0			0
$741$			17.2020i			0.631930i
$742$		0			0
$743$	0.165820			0.00608334			0.00304167	−	0.999995i	$-0.499032\pi$
$743$	0.165820			0.00608334			0.00304167	+	0.999995i	$0.499032\pi$
$744$		0			0
$745$	−3.21838			−0.117912
$746$		0			0
$747$	4.01873			0.147038
$748$		0			0
$749$	−89.0994			−3.25562
$750$		0			0
$751$	1.58336			0.0577777			0.0288889	−	0.999583i	$-0.490803\pi$
$751$	1.58336			0.0577777			0.0288889	+	0.999583i	$0.490803\pi$
$752$		0			0
$753$			57.3764i			2.09091i
$754$		0			0
$755$	−13.7808			−0.501535
$756$		0			0
$757$		−	36.6506i		−	1.33209i	−0.745912	−	0.666045i	$-0.767986\pi$
$757$		−	36.6506i		−	1.33209i	0.745912	−	0.666045i	$-0.232014\pi$
$758$		0			0
$759$	−17.7636	+	50.0686i	−0.644777	+	1.81737i
$760$		0			0
$761$	−34.4930			−1.25037			−0.625186	−	0.780476i	$-0.714977\pi$
$761$	−34.4930			−1.25037			−0.625186	+	0.780476i	$0.714977\pi$
$762$		0			0
$763$			8.77013i			0.317500i
$764$		0			0
$765$	−4.14480			−0.149855
$766$		0			0
$767$		−	37.2569i		−	1.34527i
$768$		0			0
$769$		−	0.946231i		−	0.0341220i	−0.999854	−	0.0170610i	$-0.994569\pi$
$769$		−	0.946231i		−	0.0341220i	0.999854	−	0.0170610i	$-0.00543094\pi$
$770$		0			0
$771$			7.47566i			0.269229i
$772$		0			0
$773$		−	4.34100i		−	0.156135i	−0.996948	−	0.0780674i	$-0.975125\pi$
$773$		−	4.34100i		−	0.156135i	0.996948	−	0.0780674i	$-0.0248749\pi$
$774$		0			0
$775$		−	8.00785i		−	0.287650i
$776$		0			0
$777$	−5.67593			−0.203623
$778$		0			0
$779$	−15.4787			−0.554581
$780$		0			0
$781$			68.9233i			2.46627i
$782$		0			0
$783$			15.3322i			0.547928i
$784$		0			0
$785$	17.1598			0.612459
$786$		0			0
$787$	−4.11417			−0.146654			−0.0733272	−	0.997308i	$-0.523362\pi$
$787$	−4.11417			−0.146654			−0.0733272	+	0.997308i	$0.523362\pi$
$788$		0			0
$789$		−	29.9343i		−	1.06569i
$790$		0			0
$791$		−	33.7267i		−	1.19918i
$792$		0			0
$793$			22.8026i			0.809744i
$794$		0			0
$795$		−	8.36512i		−	0.296680i
$796$		0			0
$797$			37.5860i			1.33137i	0.746235	+	0.665683i	$0.231860\pi$
$797$			37.5860i			1.33137i	−0.746235	+	0.665683i	$0.768140\pi$
$798$		0			0
$799$	28.4195			1.00541
$800$		0			0
$801$		−	7.08285i		−	0.250260i
$802$		0			0
$803$	−88.8570			−3.13569
$804$		0			0
$805$	7.04097	−	19.8458i	0.248162	−	0.699471i
$806$		0			0
$807$		−	17.4267i		−	0.613449i
$808$		0			0
$809$	15.0464			0.529004			0.264502	−	0.964385i	$-0.414792\pi$
$809$	15.0464			0.529004			0.264502	+	0.964385i	$0.414792\pi$
$810$		0			0
$811$		−	31.9509i		−	1.12195i	−0.827834	−	0.560974i	$-0.810427\pi$
$811$		−	31.9509i		−	1.12195i	0.827834	−	0.560974i	$-0.189573\pi$
$812$		0			0
$813$	15.8771			0.556833
$814$		0			0
$815$	9.39428			0.329067
$816$		0			0
$817$	−6.18880			−0.216519
$818$		0			0
$819$	−11.2293			−0.392385
$820$		0			0
$821$	51.8133			1.80830			0.904148	−	0.427219i	$-0.140507\pi$
$821$	51.8133			1.80830			0.904148	+	0.427219i	$0.140507\pi$
$822$		0			0
$823$			10.7878i			0.376038i	0.982165	+	0.188019i	$0.0602067\pi$
$823$			10.7878i			0.376038i	−0.982165	+	0.188019i	$0.939793\pi$
$824$		0			0
$825$			11.0776i			0.385673i
$826$		0			0
$827$	6.32420			0.219914			0.109957	−	0.993936i	$-0.464929\pi$
$827$	6.32420			0.219914			0.109957	+	0.993936i	$0.464929\pi$
$828$		0			0
$829$	3.09251			0.107407			0.0537036	−	0.998557i	$-0.482897\pi$
$829$	3.09251			0.107407			0.0537036	+	0.998557i	$0.482897\pi$
$830$		0			0
$831$			0.0807164i			0.00280002i
$832$		0			0
$833$		−	85.4999i		−	2.96240i
$834$		0			0
$835$	14.4213			0.499069
$836$		0			0
$837$	36.5130			1.26207
$838$		0			0
$839$	1.43324			0.0494810			0.0247405	−	0.999694i	$-0.492124\pi$
$839$	1.43324			0.0494810			0.0247405	+	0.999694i	$0.492124\pi$
$840$		0			0
$841$	−17.6930			−0.610105
$842$		0			0
$843$	−8.07979			−0.278283
$844$		0			0
$845$			5.45750i			0.187744i
$846$		0			0
$847$	101.568			3.48993
$848$		0			0
$849$			34.8831i			1.19719i
$850$		0			0
$851$	−1.09322	+	3.08136i	−0.0374751	+	0.105628i
$852$		0			0
$853$	−15.9933			−0.547601			−0.273801	−	0.961786i	$-0.588281\pi$
$853$	−15.9933			−0.547601			−0.273801	+	0.961786i	$0.588281\pi$
$854$		0			0
$855$			1.25703i			0.0429894i
$856$		0			0
$857$	4.60500			0.157304			0.0786518	−	0.996902i	$-0.474938\pi$
$857$	4.60500			0.157304			0.0786518	+	0.996902i	$0.474938\pi$
$858$		0			0
$859$		−	40.0694i		−	1.36715i	−0.729880	−	0.683575i	$-0.760424\pi$
$859$		−	40.0694i		−	1.36715i	0.729880	−	0.683575i	$-0.239576\pi$
$860$		0			0
$861$		−	61.0272i		−	2.07980i
$862$		0			0
$863$		−	34.8366i		−	1.18585i	−0.805257	−	0.592925i	$-0.797973\pi$
$863$		−	34.8366i		−	1.18585i	0.805257	−	0.592925i	$-0.202027\pi$
$864$		0			0
$865$		−	3.26360i		−	0.110966i
$866$		0			0
$867$			59.6910i			2.02721i
$868$		0			0
$869$	−89.8216			−3.04699
$870$		0			0
$871$	34.5822			1.17177
$872$		0			0
$873$		−	9.10571i		−	0.308182i
$874$		0			0
$875$		−	4.39085i		−	0.148438i
$876$		0			0
$877$	−17.6284			−0.595267			−0.297634	−	0.954680i	$-0.596197\pi$
$877$	−17.6284			−0.595267			−0.297634	+	0.954680i	$0.596197\pi$
$878$		0			0
$879$	41.3536			1.39482
$880$		0			0
$881$			24.8605i			0.837573i	0.908085	+	0.418786i	$0.137544\pi$
$881$			24.8605i			0.837573i	−0.908085	+	0.418786i	$0.862456\pi$
$882$		0			0
$883$			32.8068i			1.10404i	0.833832	+	0.552018i	$0.186142\pi$
$883$			32.8068i			1.10404i	−0.833832	+	0.552018i	$0.813858\pi$
$884$		0			0
$885$		−	16.4432i		−	0.552733i
$886$		0			0
$887$		−	26.1630i		−	0.878466i	−0.898373	−	0.439233i	$-0.855250\pi$
$887$		−	26.1630i		−	0.878466i	0.898373	−	0.439233i	$-0.144750\pi$
$888$		0			0
$889$			32.6527i			1.09514i
$890$		0			0
$891$	−60.9433			−2.04168
$892$		0			0
$893$		−	8.61902i		−	0.288425i
$894$		0			0
$895$	−22.9296			−0.766451
$896$		0			0
$897$	−13.0628	+	36.8190i	−0.436155	+	1.22935i
$898$		0			0
$899$		−	26.9271i		−	0.898068i
$900$		0			0
$901$	−30.7178			−1.02336
$902$		0			0
$903$		−	24.4003i		−	0.811993i
$904$		0			0
$905$	9.39717			0.312373
$906$		0			0
$907$	41.7791			1.38725			0.693627	−	0.720335i	$-0.256012\pi$
$907$	41.7791			1.38725			0.693627	+	0.720335i	$0.256012\pi$
$908$		0			0
$909$	−11.0453			−0.366350
$910$		0			0
$911$	−22.8227			−0.756150			−0.378075	−	0.925775i	$-0.623414\pi$
$911$	−22.8227			−0.756150			−0.378075	+	0.925775i	$0.623414\pi$
$912$		0			0
$913$	−39.4411			−1.30531
$914$		0			0
$915$			10.0638i			0.332700i
$916$		0			0
$917$		−	60.4852i		−	1.99740i
$918$		0			0
$919$	44.1825			1.45745			0.728723	−	0.684809i	$-0.240114\pi$
$919$	44.1825			1.45745			0.728723	+	0.684809i	$0.240114\pi$
$920$		0			0
$921$	23.5523			0.776076
$922$		0			0
$923$			50.6842i			1.66829i
$924$		0			0
$925$			0.681746i			0.0224157i
$926$		0			0
$927$	2.61676			0.0859456
$928$		0			0
$929$	−25.5961			−0.839780			−0.419890	−	0.907575i	$-0.637931\pi$
$929$	−25.5961			−0.839780			−0.419890	+	0.907575i	$0.637931\pi$
$930$		0			0
$931$	−25.9303			−0.849830
$932$		0			0
$933$	−30.0248			−0.982967
$934$		0			0
$935$	40.6784			1.33033
$936$		0			0
$937$		−	5.33194i		−	0.174187i	−0.996200	−	0.0870935i	$-0.972242\pi$
$937$		−	5.33194i		−	0.174187i	0.996200	−	0.0870935i	$-0.0277579\pi$
$938$		0			0
$939$	−3.69096			−0.120450
$940$		0			0
$941$			17.8305i			0.581256i	0.956836	+	0.290628i	$0.0938643\pi$
$941$			17.8305i			0.581256i	−0.956836	+	0.290628i	$0.906136\pi$
$942$		0			0
$943$	−33.1305	−	11.7542i	−1.07888	−	0.382769i
$944$		0			0
$945$	20.0207			0.651274
$946$		0			0
$947$		−	19.2271i		−	0.624798i	−0.949951	−	0.312399i	$-0.898867\pi$
$947$		−	19.2271i		−	0.624798i	0.949951	−	0.312399i	$-0.101133\pi$
$948$		0			0
$949$	−65.3429			−2.12112
$950$		0			0
$951$		−	34.3697i		−	1.11451i
$952$		0			0
$953$		−	37.9108i		−	1.22805i	−0.789287	−	0.614025i	$-0.789549\pi$
$953$		−	37.9108i		−	1.22805i	0.789287	−	0.614025i	$-0.210451\pi$
$954$		0			0
$955$			4.96782i			0.160755i
$956$		0			0
$957$			37.2494i			1.20410i
$958$		0			0
$959$			1.48296i			0.0478873i
$960$		0			0
$961$	−33.1256			−1.06857
$962$		0			0
$963$	12.0794			0.389254
$964$		0			0
$965$		−	16.7773i		−	0.540080i
$966$		0			0
$967$			11.2963i			0.363264i	0.983367	+	0.181632i	$0.0581380\pi$
$967$			11.2963i			0.363264i	−0.983367	+	0.181632i	$0.941862\pi$
$968$		0			0
$969$	27.8789			0.895598
$970$		0			0
$971$	−19.3047			−0.619516			−0.309758	−	0.950815i	$-0.600248\pi$
$971$	−19.3047			−0.619516			−0.309758	+	0.950815i	$0.600248\pi$
$972$		0			0
$973$			43.4844i			1.39405i
$974$		0			0
$975$			8.14616i			0.260886i
$976$		0			0
$977$		−	0.503192i		−	0.0160985i	−0.999968	−	0.00804926i	$-0.997438\pi$
$977$		−	0.503192i		−	0.0160985i	0.999968	−	0.00804926i	$-0.00256219\pi$
$978$		0			0
$979$			69.5134i			2.22166i
$980$		0			0
$981$		−	1.18899i		−	0.0379614i
$982$		0			0
$983$	−26.6253			−0.849216			−0.424608	−	0.905377i	$-0.639588\pi$
$983$	−26.6253			−0.849216			−0.424608	+	0.905377i	$0.639588\pi$
$984$		0			0
$985$		−	0.153918i		−	0.00490424i
$986$		0			0
$987$	33.9819			1.08166
$988$		0			0
$989$	−13.2465	−	4.69965i	−0.421214	−	0.149440i
$990$		0			0
$991$			46.9800i			1.49237i	0.665740	+	0.746184i	$0.268116\pi$
$991$			46.9800i			1.49237i	−0.665740	+	0.746184i	$0.731884\pi$
$992$		0			0
$993$	58.1564			1.84554
$994$		0			0
$995$		−	20.4354i		−	0.647845i
$996$		0			0
$997$	−59.9944			−1.90004			−0.950020	−	0.312188i	$-0.898938\pi$
$997$	−59.9944			−1.90004			−0.950020	+	0.312188i	$0.898938\pi$
$998$		0			0
$999$	−3.10852			−0.0983492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.i.b.1471.12	yes	48
4.3	odd	2		3680.2.i.a.1471.37	yes	48
23.22	odd	2		3680.2.i.a.1471.12	✓	48
92.91	even	2	inner	3680.2.i.b.1471.37	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.i.a.1471.12	✓	48	23.22	odd	2
3680.2.i.a.1471.37	yes	48	4.3	odd	2
3680.2.i.b.1471.12	yes	48	1.1	even	1	trivial
3680.2.i.b.1471.37	yes	48	92.91	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.89612i q^{3} +1.00000i q^{5} +4.39085 q^{7} -0.595278 q^{9} +O(q^{10})\)	\(q-1.89612i q^{3} +1.00000i q^{5} +4.39085 q^{7} -0.595278 q^{9} +5.84225 q^{11} +4.29622 q^{13} +1.89612 q^{15} -6.96280i q^{17} -2.11167 q^{19} -8.32558i q^{21} +(-4.51980 - 1.60356i) q^{23} -1.00000 q^{25} -4.55965i q^{27} -3.36258 q^{29} +8.00785i q^{31} -11.0776i q^{33} +4.39085i q^{35} -0.681746i q^{37} -8.14616i q^{39} +7.33008 q^{41} +2.93077 q^{43} -0.595278i q^{45} +4.08162i q^{47} +12.2795 q^{49} -13.2023 q^{51} -4.41170i q^{53} +5.84225i q^{55} +4.00398i q^{57} -8.67203i q^{59} +5.30759i q^{61} -2.61377 q^{63} +4.29622i q^{65} +8.04944 q^{67} +(-3.04054 + 8.57009i) q^{69} +11.7974i q^{71} -15.2094 q^{73} +1.89612i q^{75} +25.6524 q^{77} -15.3745 q^{79} -10.4315 q^{81} -6.75102 q^{83} +6.96280 q^{85} +6.37587i q^{87} +11.8984i q^{89} +18.8640 q^{91} +15.1839 q^{93} -2.11167i q^{95} +15.2966i q^{97} -3.47776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 8 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q + 8 * q^7 - 48 * q^9	\( 48 q + 8 q^{7} - 48 q^{9} + 16 q^{11} + 20 q^{23} - 48 q^{25} - 8 q^{29} + 8 q^{41} + 56 q^{49} - 24 q^{51} - 120 q^{63} - 32 q^{67} - 20 q^{69} + 32 q^{77} + 72 q^{81} + 64 q^{83} + 40 q^{91} - 32 q^{93} - 144 q^{99}+O(q^{100}) \) 48 * q + 8 * q^7 - 48 * q^9 + 16 * q^11 + 20 * q^23 - 48 * q^25 - 8 * q^29 + 8 * q^41 + 56 * q^49 - 24 * q^51 - 120 * q^63 - 32 * q^67 - 20 * q^69 + 32 * q^77 + 72 * q^81 + 64 * q^83 + 40 * q^91 - 32 * q^93 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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